Repunit

Definition:

http://en.wikipedia.org/wiki/Repunit

http://mathworld.wolfram.com/Repunit.html

http://primes.utm.edu/glossary/page.php?sort=Repunit

Mersenne Number

http://mathworld.wolfram.com/MersenneNumber.html

Puzzles :

(1)

Prove that the number 1111 ... 11, consisting of 91 ones, is a composite number

(2)

A Mersenne prime is a number of the form 111 ... 111 (base 2).
Show that any such prime is a divisor of some number 111 ... 111 (base 10)

(3)

(a) The 12-digit number 444444444443 is prime.

Find another prime of the type a(R12) - 1,

where R12 is 111111111111, a repunit

(b) The 12-digit number 101010101011 is prime.

Find other primes of the form a(101010101010) + 1

(c) The twin numbers (383838 - 1), (383838 + 1) are primes.
Find other twins of the form a(101010) - 1, a(101010) + 1,

where a is a 2-digit number.

(d) Find twin primes of the form a(1010101) - 1, a(1010101) + 1